When doing calculus with respect to a non-Archimedian metric (such as the p-adic metric), is it true that differentiability does NOT imply continuity (example: the p-adic norm seems to be differentiable but not continuous)?
I spoke too soon. The p-adic norm was just a poor example. Instead, consider defined by (where is the set of p-adic numbers, and is the p-adic norm of ).
First, I will consider continuity at 0. Consider the sequence . This sequence converges to 0 with respect to the p-adic metric, but for all while . Hence, is not continuous at . Yet:
Hence, the derivative exists at . So, differentiability does not imply continuity. Did I make a mistake somewhere?
Edit: In general, if , then if , , so
Hence, agrees with the derivative found for , as well.